We have investigated the dynamics of floating tracer in an idealised turbulent quasi-geostrophic ocean by advecting Lagrangian particles in a high-resolution velocity field enhanced by the potential flow associated with vortex stretching. At first order in the Rossby number expansion, this component of the ageostrophic circulation can be derived through a diagnostic equation in terms of the geostrophic velocities. Borrowing methods from the theory of Lagrangian coherent structures, we identify coherent material loops around strong vortex cores using the Lagrangian averaged vorticity deviation (LAVD). Building on studies of clustering in kinematic, stochastic velocity fields, we utilise methods from statistical topography to show that the coherent vortices dominate the distribution of extreme values of the concentration field. We find that the presence of clusters and voids in a coherent vortex depends on more than just the sense of rotation, but also on the full evolution of the vorticity over its lifecycle. We identify the mechanism behind the cluster formation that respects the symmetries of the quasi-geostrophic equations but can be expected to hold robustly in more complicated regimes, due to the simple physical description. The association of cluster formation with vortex stretching implies that LAVD is a particularly relevant metric for floating tracer dynamics. The detection of intense clustering also has implications for reaction rates between ocean-borne flotsam, meaning that our results are relevant to understanding the general risk of floating microplastics and marine biological populations.

We develop a time-dependent conformal method to study the effect of viscosity on steep surface waves. When the effect of surface tension is included, numerical solutions are found that contain highly oscillatory parasitic capillary ripples. These small-amplitude ripples are associated with the high curvature at the crest of the underlying viscous-gravity wave, and display asymmetry about the wave crest. Previous inviscid studies of steep surface waves have calculated intricate bifurcation structures that appear for small surface tension. We show numerically that viscosity suppresses these. While the discrete solution branches still appear, they collapse to form a single smooth branch in the limit of small surface tension. These solutions are shown to be temporally stable, both to small superharmonic perturbations in a linear stability analysis, and to some larger amplitude perturbations in different initial-value problems. Our work provides a convenient method for the numerical computation and analysis of water waves with viscosity, without evaluating the free-boundary problem for the full Navier–Stokes equations, which becomes increasingly challenging at larger Reynolds numbers.

Gravito–capillary waves at free surfaces are ubiquitous in several natural and industrial processes involving quiescent liquid pools bounded by cylindrical walls. These waves emanate from the relaxation of initial interface distortions, which often take the form of a cavity (depression) centred on the symmetry axis of the container. The surface waves reflect from the container walls leading to a radially inward propagating wavetrain converging (focussing) onto the symmetry axis. Under the inviscid approximation and for sufficiently shallow cavities, the relaxation is well-described by the linearised potential-flow equations. Naturally, adding viscosity to such a system introduces viscous dissipation that enervates energy and dampens the oscillations at the symmetry axis. However, for viscous liquids and deeper cavities, these equations are qualitatively inaccurate. In this study, we decompose the initial localised interface distortion into several Bessel functions and study their time evolution governing the propagation of concentric gravito–capillary waves on a free surface. This is carried out for inviscid as well as viscous liquids. For a sufficiently deep cavity, the inward focussing of waves results in large interfacial oscillations at the axis, necessitating a second-order nonlinear theory. We demonstrate that this theory effectively models the interfacial behaviour and highlights the crucial role of nonlinearity near the symmetry axis. This is rationalised via demonstration of the contribution of bound wave components to the interface displacement at the symmetry axis Contrary to expectations, the addition of slight viscosity further intensifies the oscillations at the symmetry axis although the mechanism of wavetrain generation here is quite different compared with bubble bursting where such behaviour is well known (Duchemin et al., Phys. Fluids, vol. 14, issue 9, 2002, pp. 3000–3008). This finding underscores the limitations of the potential flow model and suggests avenues for more accurate modelling of such complex free-surface flows.

Aircraft with bio-inspired flapping wings that are operated in low-density atmospheric environments encounter unique challenges associated with the low density. The low density results in the requirement of high operating velocities of aircraft to generate sufficient lift resulting in significant compressibility effects. Here, we perform numerical simulations to investigate the compressibility effects on the lift generation of a bio-inspired wing during hovering flight using an immersed boundary method. The aim of this study is to develop a scaling law to understand how the lift is influenced by the Reynolds and Mach numbers, and the associated flow physics. Our simulations have identified a critical Mach number of approximately $0.6$ defined by the average wing-tip velocity. When the Mach number is lower than 0.6, compressibility does not have significant effects on the lift or flow fields, while when the Mach number is greater than $0.6$, the lift coefficient decreases linearly with increasing Mach number, due to the drastic change in the pressure on the wing surface caused by unsteady shock waves. Moreover, the decay rate is dependent on the Reynolds number and the angle of attack. Based on these observations, we propose a scaling law for the lift of a hovering flapping wing by considering compressible and viscous effects, with the scaled lift showing excellent collapse.

Models for slow flow of dense granular materials often treat the medium as incompressible, thereby neglecting the role of Reynolds dilatancy. However, recent particle simulations have demonstrated the presence of a significant coupling between the volume fraction and velocity fields. The model of Dsouza & Nott (J. Fluid Mech., vol. 888, 2020, R3) incorporates dilatancy and captures the coupling, but it has thus far lacked experimental validation. In this paper, we provide the first experimental demonstration of dilatancy and its coupling to the kinematics in a two-dimensional cylindrical Couette cell. We find a shear layer near the inner cylinder within which there is significant dilation. Within the shear layer, the azimuthal velocity decays roughly exponentially and the volume fraction rises with radial distance from the inner cylinder. The predictions of the model of Dsouza & Nott (2020) are in good agreement with the experimental data for a variety of roughness features of the outer cylinder. Moreover, by comparing the steady states resulting from different initial volume fraction profiles (but having the same average), we show the inter-dependence of the velocity and volume fraction fields, as predicted by the model. Our results establish the importance of shear dilatancy even in systems of constant volume.

In this study, we investigate the properties of energy thickness $\delta _3$ in turbulent boundary layer (TBL) flows, a parameter derived solely from the mean streamwise velocity ($U$) profile. Through an analysis of the energy integral equation for zero pressure gradient TBLs, we establish a close relationship between turbulent kinetic energy (TKE) production and $\delta _3$, offering a practical method to estimate TKE production, which is particularly useful in physical experiments where direct measurements are challenging. The significance of $\delta _3$ becomes even more pronounced in TBLs under pressure gradient. Through extensive analysis of numerical and experimental data, we show that the ratio between $\delta _3$ and the momentum thickness $\delta _2$ is a promising criterion for predicting flow separation. Moreover, we derive a new energy integral equation for TBLs under arbitrary pressure gradients, and provide approximations for TKE productions terms by $R_{uv}\,\partial U/\partial y$ and $R_{uu}\,\partial U/\partial x$, and dissipation term by the mean shear. Here, $x, y$ represent the streamwise and wall-normal directions, respectively, and $R_{uu}$ and $R_{uv}$ are the Reynolds normal and shear stresses. The accuracy and robustness of the new energy integral equation and the approximation equations are validated using direct numerical simulations data. Our results show that the TKE production by $R_{uv}\,\partial U/\partial y$ and the overall productions consistently remain positive, reflecting a continuous conversion of mean kinetic energy into TKE across all TBLs. However, under strong favourable pressure gradients, TKE production by $R_{uu}\,\partial U/\partial x$ becomes negative, indicating a reverse energy transfer from TKE to mean kinetic energy.

Predicting and perhaps mitigating against rare, extreme events in fluid flows is an important challenge. Due to the time-localised nature of these events, Fourier-based methods prove inefficient in capturing them. Instead, this paper uses wavelet-based methods to understand the underlying patterns in a forced flow over a 2-torus which has intermittent high-energy burst events interrupting an ambient low-energy ‘quiet’ flow. Two wavelet-based methods are examined to predict burst events: (i) a wavelet proper orthogonal decomposition (WPOD) based method which uncovers and utilises the key flow patterns seen in the quiet regions and the bursting episodes; and (ii) a wavelet resolvent analysis (WRA) based method that relies on the forcing structures which amplify the underlying flow patterns. These methods are compared with a straightforward energy tracking approach which acts as a benchmark. Both the wavelet-based approaches succeed in producing better predictions than a simple energy criterion, i.e. earlier prediction times and/or fewer false positives and the WRA-based technique always performs better than WPOD. However, the improvement of WRA over WPOD is not as substantial as anticipated. We conjecture that this is because the mechanism for the bursts in the flow studied is found to be largely modal, associated with the unstable eigenfunction of the Navier–Stokes operator linearised around the mean flow. The WRA approach should deliver much better improvement over the WPOD approach for generically non-modal bursting mechanisms where there is a lag between the imposed forcing and the final response pattern.

The three-dimensional stability of two-dimensional natural convection flows in a heated, square enclosure inclined to the horizontal is investigated numerically. First, the computational procedure is validated by comparison of base flow solutions to results reported in literature across a range of inclinations. A bi-global linear stability analysis is then conducted to investigate the stability of these two-dimensional base flows to infinitesimal three-dimensional perturbations, and the effect that buoyancy forces (defined by a buoyancy number $R_N$) and enclosure inclination $\theta$ have on these stability characteristics. The flow is first observed to become three-dimensionally unstable at buoyancy number $R_N = 213.8$ when $\theta$ is $180^\circ$; this increases to $R_N = 2.54 \times 10^4$ at inclination $\theta =58^\circ$. It is found that the two-dimensional base flow is more unstable to three-dimensional perturbations with the critical $R_N$ corresponding to three-dimensional instability being significantly lower than its two-dimensional counterpart across all considered inclinations except $83^\circ \leq \theta \leq 88^\circ$, where the most unstable mode is a two-dimensional oscillatory mode that develops in the boundary layers along the conducting walls. Eight different leading three-dimensional instability modes are identified, with inclinations $58^\circ \leq \theta < 88^\circ$ transitioning through an oscillatory mode, and inclinations $88^\circ \leq \theta \leq 180^\circ$ transitioning through a stationary mode. The characteristics of the primary instability modes corresponding to inclinations $88^\circ \leq \theta \leq 179^\circ$ indicate the presence of a Taylor–Görtler instability.

We undertake an experimental investigation into the instabilities that emerge when a shear-thinning fluid intrudes a less viscous Newtonian fluid axisymmetrically in a lubricated Hele-Shaw cell. Pre-formed lubrication layers of Newtonian fluid that separate the shear-thinning fluid from the cell walls are incorporated into the experimental design. Provided the lubrication layers remain effective at reducing shear stress, so that extensional stresses dominate the flow of the intruding fluid, the instabilities evolve to form branch-like structures, which exhibit fracturing or tearing behaviour at their troughs. Thicker lubrication layers enable the branches to propagate radially outwards, whilst thinner, less effective ones hinder their development and progression. In the absence of lubrication layers, the shear-thinning fluid spreads radially and remains axisymmetric. For lubricated flows, we show that the number of branches is dependent primarily on the strain rate at the radial distance where they first emerge, and that the number of branches decreases with increasing strain rate.

The manipulation of the Richtmyer–Meshkov instability growth at a heavy–light interface via successive shocks is theoretically analysed and experimentally realized in a specific shock-tube facility. An analytical model is developed to forecast the interface evolution before and after the second shock impact, and the possibilities for the amplitude evolution pattern are systematically discussed. Based on the model, the parameter conditions for each scenario are designed, and all possibilities are experimentally realized by altering the time interval between two shock impacts. These findings may enhance the understanding of how successive shocks influence hydrodynamic instabilities in practical applications.

Mangroves are a natural defence of the coastal strip against extreme waves. Furthermore, innovative techniques of naturally based coast defence are used increasingly, according to the canons of eco-hydraulics. Therefore, it is important to correctly evaluate the transmission of waves through cylinder arrays. In the present paper, the attenuation of solitary waves propagating through an array of rigid emergent and submerged cylindrical stems on a horizontal bottom is investigated theoretically, numerically and experimentally. The results of the theoretical model are compared with the numerical simulations obtained with the smoothed particle hydrodynamics meshless Lagrangian numerical code and with experimental laboratory data. In the latter case, solitary waves were tested on a background current, in order to reproduce more realistic sea conditions, since the absence of circulation currents is very rare in the sea. The comparison confirmed the validity of the theoretical model, allowing its use for the purposes indicated above. Furthermore, the present study allowed for an evaluation of the bulk drag coefficient of the rigid stem arrays used, as a function of their density, the stem diameter, and their submergence ratio.

Numerical simulations are conducted to investigate particle suspension and deposition within turbidity currents. Utilizing Lagrangian particle tracking and a discrete element model, our numerical approach enables a detailed examination of autosuspension, deposition and bulk behaviours of turbidity current. We specifically focus on flow regimes where particle settling and buoyancy-induced hydrodynamics play equally important roles. Our discussion is divided into three parts. Firstly, we examine the main body of the current formed by suspended particles, revealing a temporal evolution consisting of initial slumping, propagation and dissipation stages. Our particle calculation allows for the tracking of autosuspended particles, enabling a deeper understanding of the connection between autosuspension and current propagation through energy budget analysis. In the second part, we delve into particle deposition, highlighting transverse and longitudinal variations. Transverse variations arise from lobe-and-cleft (LC) flow features, while longitudinal variations result from vortex detachment, particularly notable with large-sized particles. We observe that as particle size increases, leading to a particle Stokes number greater than 0.1, rapid particle settling suppresses the LC flow structure, resulting in wider lobes at the deposition height. Lastly, we propose a new scaling law for the propagation speed and current length. Our simulation results demonstrate close agreement with this new scaling law, providing valuable insights into turbidity current dynamics.

The collapse of an initially spherical cavitation bubble near a free surface leads to the formation of two jets: a downward jet into the liquid, and an upward jet penetrating the free surface. In this study, we examine the surprising interaction of a bubble trapped in a stable cavitating vortex ring approaching a free surface. As a result, a single fast and tall liquid jet forms. We find that this jet is observed only above critical Froude numbers ($Fr$) and Weber numbers ($We$) when ${Fr}^2 (1.6-2.73/{We}) > 1$, illustrating the importance of inertia, gravity and surface tension in accelerating this novel jet and thereby reaching heights several hundred times the radius of the vortex ring. Our experimental results are supported by numerical simulations, revealing that the underlying mechanism driving the vortex ring acceleration is the disruption of the equilibrium of high-pressure regions at the front and rear of the vortex ring caused by the free surface. Quantitative analysis based on the energy relationships elucidates that the velocity ratio between the maximum velocity of the free-surface jet and the translational velocity of the vortex ring is relatively stable yet is attenuated by surface tension when the jet is mild.

We investigate the drag reduction effect of the streamwise travelling wave-like wall deformation in a high-Reynolds-number turbulent channel flow by large-eddy simulation (LES). First, we assess the validity of subgrid-scale models in uncontrolled and controlled flows. For friction Reynolds numbers $Re_\tau = 360$ and $720$, the Smagorinsky and wall-adapting local eddy-viscosity (WALE) models with a damping function can reproduce well the mean velocity profile obtained by direct numerical simulation (DNS) in both the uncontrolled and controlled flows, leading to a small difference in drag reduction rate between LES and DNS. The LES with finer grid resolution can reproduce well the key structures observed in the DNS of the controlled flow. These results show that the high-fidelity LES is valid for appropriately predicting the drag reduction effect. In addition, a small computational domain is sufficient for reproducing the turbulence statistics, key structures and drag reduction rate obtained by DNS. Subsequently, to investigate the trend of drag reduction rate at higher Reynolds numbers, we utilize the WALE model with the damping function to investigate the control effect at higher Reynolds numbers up to $Re_\tau = 3240$. According to the analyses of turbulence statistics and instantaneous flow fields, the drag reduction at higher Reynolds numbers occurs basically through the same mechanism as that at lower Reynolds numbers. In addition, the drag reduction rate obtained by the present LES approaches that predicted using the semi-empirical formula (Nabae et al., Intl J. Heat Fluid Flow, vol. 82, 2020, 108550) as the friction Reynolds number increases, which supports the high predictability of the semi-empirical formula at significantly high Reynolds numbers.